nach oben

2008 | Buch

Kapitel lesen Erstes Kapitel lesen

Absolute Stability of Nonlinear Control Systems

verfasst von: Xiaoxin Liao, Pei Yu

Verlag: Springer Netherlands

Buchreihe : Mathematical Modelling: Theory and Applications

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Einloggen, um Zugang zu erhalten

Über dieses Buch

Following the recent developments in the field of absolute stability, Prof. Xiaoxin Liao, in conjunction with Prof. Pei Yu, has created a second edition of his seminal work on the subject. Liao begins with an introduction to the Lurie problem and Lurie control system, before moving on to the simple algebraic sufficient conditions for the absolute stability of autonomous and non-autonomous ODE systems, as well as several special classes of Lurie-type systems. The focus of the book then shifts toward the new results and research that have appeared in the decade since the first edition was published. This book is aimed to be used by undergraduates in the areas of applied mathematics, nonlinear control systems, and chaos control and synchronisation, but may also be useful as a reference for researchers and engineers. The book is self-contained, though a basic knowledge of calculus, linear system and matrix theory, and ordinary differential equations is a prerequisite.

Inhaltsverzeichnis

Frontmatter

1. Introduction

As we know, the stability is the most fundamental problem in the design of automatic control systems, since only a stable system can keep working properly under disturbances [3, 4, 6, 8–10]. In fact, automatic control theory began from Maxwell’s study on the stability of Watt centrifugal governor. When one designs a control system, one first needs to consider some type of stability for the system and then investigate other problems.

Among various stability theories, the Lyapunov stability is still the most important one [56–63, 101–109, 128, 164, 186, 187]. However, the main difficulty in analyzing Lyapunov stability is how to determine a Lyapunov function for a given system. There does not exist general rules for constructing Lyapunov functions, but are merely based on a researcher or designer’s experience and some particular techniques. The first-order approximation method and many results obtained for the first and second critical cases demand very restrictive requirements on nonlinear terms, which cause difficulties in applications. Moreover, it should be pointed out that the Lyapunov stability theory is mainly applicable for local stability, while many practical problems need to consider globally asymptotic stability or even globally exponential stability.

2. Principal Theorems on Global Stability

In this chapter, we introduce the main tools and principal results that play fundamental roles for the whole book, such as Lyapunov function, K-class function (or wedge function), Dini-derivative, M-matrix, Hurwitz matrix, positive (negative) definite matrix; and the principal theorems on global stability, partial global stability, and global stability of sets.

Partial materials presented in this chapter are due to Lyapunov [97], Hahn [38], Malkin [102] (Sect. 2.1), Yoshizawa [170] (Sect. 2.2), Rumyantsev [133, 134] (Sect. 2.4.2), Liao [68, 69] (Sect. 2.3-2.6), and Liao [69, 70] (Sect. 2.7 and 2.8).

3. Sufficient Conditions of Absolute Stability: Classical Methods

The development of control theory began from Maxwell’s study on the stability of Watt centrifugal governor. In early 1940s, the former Soviet Union scholars Lurie, Postnikov, and others developed a method to deal with a class of nonlinear systems, now called nonlinear isolate method. Lurie and his coworkers studied many real control systems, including the Bulgakov problem of aircraft automatic control. They first isolated the nonlinear part from the system and considered it as a feedback control of the system so that the system has a closed-loop form. This problem is the well-known Lurie problem [99, 100]. The research on Lurie’s problem has so far resulted in a number of monographs and several hundred of scientific publications. In fact, the pose of the Lurie problem actually initiated the research on the robust control and robust stability for nondeterministic systems or multivalued differential equations. It promoted the application and development of stability theory. In recent years, the developments in chaos control and chaos synchronization, neural networks, electrical systems have revealed that these research areas are closely related to Lurie control systems. Thus, it is useful and important to further study Lurie control systems, in particular, from the point of control theory and stability theory. [164, 173–179]

In this chapter, we briefly introduce the background of Lurie problem, the mathematical methodology for solving Lurie problem, and consider three classical methods for studying absolute stability: the Lyapunov–LurieV-function method (having integrals and quadratic form), S-program, and Popov frequency-domain criterion and simplified Popov method.We also consider some equivalent conditions at which the derivative of the Lyapunov-LurieV-function is negative definite.

Some materials are based on the results of Liao [68] (Sects. 3.1 and 3.2), Lurie [99, 100] (Sect. 3.3), Zhu [184] and Zhao [180] (Sect. 3.4), as well as Popov [120– 123] and Zhang [178] (Sect. 3.5).

4. Necessary and Sufficient Conditions for Absolute Stability

In this chapter, we discuss the necessary and sufficient conditions for absolute stability of various Lurie control systems described by ordinary differential equations. The absolute stability for all the system’s variables will be equivalently transformed into that of a single variable or partial variables, and that of the Hurwitz stability for matrix, which is easy to be verified. Based on obtained theoretical results, some practically useful algebraic sufficient conditions will be derived, which provide guidelines for designers and engineers. The material given in 4.1 and 4.3 is chosen from [78, 80] and that presented in 4.2 and 4.4 is based on [67, 76, 77]. The results given in 4.5 are mainly taken from [89].

5. Special Lurie-Type Control Systems

In this chapter, we present some necessary and sufficient algebraic conditions for the absolute stability of several special classes of Lurie-type control systems. Moreover, the algebraic sufficient conditions for absolute stability of other systems are obtained. All conditions are convenient in applications, in particular, for designing absolute stable control systems, or for stabilizing nonabsolute stable control systems.

Part of this chapter is based on the results of Ye [163], Xie [158], and Zhang [177] (Sect. 5.1); Liao [72, 78] (Sects. 5.3–5.5); Letov [63] (Sects. 5.2 and 5.4); and Shu et al. [136] (Sect. 5.6).

6. Nonautonomous Systems

In this chapter, we consider nonautonomous systems. Lurie control systems are mainly autonomous systems. Thus, the Lurie method or Popov method was developed for single-variable autonomous systems, which are very difficult to be used to study nonautonomous systems. However, many practical systems contain time-variant parameters though they usually vary slowly. Stability of time-variant systems has been a relatively difficult problem in control systems and dynamical systems, and has less results compared to that of autonomous systems. The results presented in this chapter are mainly taken from [67, 86] for Sects. 6.1–6.4, and from [67, 105] for Sect. 6.5.

7. Systems with Multiple Nonlinear Feedback Controls

As a result of the fast development of science and technology, control systems have become more and more complex. Usually, single feedback control is not enough to finish a complicated task and needs multiple feedbacks. In 1988, SIAM published a research report “The Future Development of Control Theory – Mathematical Prospect.” It was indicated in this report that though the stability of nonlinear control systems had been paid much attention and many mathematical results had been found, the results are still mainly on single-variable nonlinear control systems such as Popov’s principal and Lyapunov method. Multivariable nonlinear control systems are not yet well generalized. It is still difficult to extend the single-variable case to the multivariable case.

In this chapter, we will discuss the absolute stability of control systems with multiple nonlinear control terms. The results given in Sects. 7.1–7.4 are taken from Liao et al. [91, 92], and that presented in Sect. 7.5 are from Gan and Ge [29].

8. Robust Absolute Stability of Interval Control Systems

Strictly speaking, a mathematical model is only an approximate description of a real system since the information of the system coefficients are usually the upper and lower bounds, not the exact values [44]. In the past two decades, the stability study for linear control systems with parameters varied in a finite closed interval has been a hot topic in control society. However, not many results have been obtained for stability of nonlinear control systems with varied parameters in an interval. In this chapter, we will systematically introduce robust stability of control systems with interval varied parameters. In fact, such idea and methodology can be generalized to consider other Lurie control systems. The materials presented in this chapter are chosen from Liao et al. [85] (Sects. 8.1–8.4), and from Yu and Liao [172] (Sects. 8.5–8.9) and Liao [79].

9. Discrete Control Systems

The Lurie control system and the well-known Lurie problem were originally developed for solving the nonlinear systems described by ordinary differential equations (ODE), and the most research interest and results in this area were focused on ODE systems. With the very fast development of computer systems and technology, the dynamics and asymptotic behavior of discrete systems described by difference equations (DE) play more important roles in solving practical problems [115], attracting more and more researchers [47]. The absolute stability of discrete Lurie control systems is naturally raised. However, the results obtained so far for such systems are relatively less than that of continuous Lurie control systems.

In this chapter, we will generalize the recently developed theory and methodology for continuous Lurie control systems to study discrete Lurie control systems described by difference equations. We will mainly discuss the sufficient and necessary conditions of absolute stability, and some practically useful algebraic suffi- cient conditions of absolute stability. The material of this chapter is mainly chosen from [77, 84].

10. Time-Delayed and Neutral Lurie Control Systems

In modeling natural and social phenomena, the dynamic behavior of many systems depends upon not only the current state, but also the system’s history, which is called time-delayed phenomenon. Mathematical models arising from the areas of engineering, physics, mechanics, control system, chemical reaction, biological, and medical systems always involve time delay. In particular, time delay often appears in control systems. Any system with a feedback control involves unavoidable time delay since time is needed for the system to appropriately react to the input. Therefore, studying the absolute stability of time-delayed Lurie systems is naturally important and necessary [51, 66].

In Chap. 3, we have obtained absolute stability conditions for the direct, indirect, and general controls of Lurie systems without time delay. However, not much attention has been paid to time-delayed Lurie systems (i.e., the Lurie systems described by differential difference equations (DDE)). Although many researchers are, with the aid of the Matlab software LMI, still investigating the stability of Lurie systems with or without time delay, the conditions they obtained are only sufficient.

In this chapter, based on the results we obtained in Chap. 3, we will continue to consider the absolute stability of Lurie systems with time delay. Some materials presented in this chapter are based on the results of [94] (Sects. 10.1 and 10.2), [160, 161] (Sect. 10.3), and Nain [110–112] (Sects. 10.4 and 10.5).

11. Control Systems Described by Functional Differential Equations

Control systems described by ordinary differential equations have been thoroughly studied, and the stability theory of such systems has been developed very rapidly [52]. In practice, in particular, for any automatic control problems with feedbacks, time-delay always appears in such systems. This is because the system needs time to process the information and make decision to react. Such time-delays are usually ignored in classical control theory. However, modern control theory has been paid attention to the effect of the time-delay in control systems, and some results have been obtained. Thus, in this second edition, we add the study of Lurie control systems described by differential and difference equations. From the development of mathematical theory, since differential and difference equations are special case of functional differential equations, it is natural to consider Lurie control systems described by functional differential equations [129–131]. In this chapter, we will present the results concerning such systems.

This chapter is mainly due to Somolinos [139] (Sect. 11.1), Zhao [181] (Sect. 11.2), Ruan and Wu [132] (Sect. 11.3), Chukwu [14] (for Sects. 11.4 and 11.5), and He [42] (Sect. 11.6).

12. Absolute Stability of Hopfield Neural Network

In this chapter, we first discuss the relationship between the stability of Hopfield neural network, Lyapunov stability, and the invariant principle in the sense of LaSalle. Next, we describe the connection and difference between the Hopfield neural network and the Lurie control systems with multiple nonlinear controllers. Then, we introduce the concept of absolute stability for neural networks, and present the sufficient and necessary conditions for two types of neural networks. Finally, we discuss various sufficient conditions for the absolute stability of Hopfield neural network. Partial materials are chosen Forti et al. [27, 28] and Kaskurewicz et al. [50] (Sects. 12.3 and 12.4), Liao et al. [87, 88, 90] (Sect. 12.5), and Liu [98] (Sect. 12.6).

13. Application to Chaos Control and Chaos Synchronization

Since Pecora and Carroll [117, 118] first designed an analog electrical circuit to realize chaos synchronization, many researchers have extensively studied the property of chaos synchronization and possible applications in practice. This has changed a long time viewpoint: chaos cannot be controlled, nor synchronized.

Although many results about chaos synchronization have been obtained on the basis of stability theory, general mathematical theory and methodology are still under development. Recently, Curran and Chua [20] suggested that different chaos synchronization methods should be unified to establish a fundamental mathematical theory on the basis of the absolute stability theory of Lurie control systems. The authors and their coworkers have also studied chaos synchronization following Curran and Chua’s idea [81, 82, 154]. For the Chua’s chaotic circuit, ywe have recently found that it could be transformed into a type of Lurie system, and thus the theory and methodology developed by Liao [72–77] can be used to study the synchronization of two Chua’s circuits. Chua’s circuit is the first electrical circuit to realize chaos in experiment, which exhibits very rich complex dynamical behavior, and yet has very high potential in real applications.

In this chapter, as an application, we will apply the absolute stability of Lurie control systems developed in previous chapters to study the globally exponential synchronization of two Chua’s chaotic circuits [95]. Also we propose and develop the theory and methodology of absolutely exponential stability, and investigate the global synchronization of two chaotic systems with feedback controls [81, 82, 89]. The materials presented in this chapter are mainly chosen from Liao and Yu [95] (Sects. 13.1–13.5), Liao et al. [89] (Sect. 13.6.1), and Liao et al. [84] (Sect. 13.6.2).

Backmatter

Titel: Absolute Stability of Nonlinear Control Systems
verfasst von: Xiaoxin Liao
Pei Yu
Verlag: Springer Netherlands
Electronic ISBN: 978-1-4020-8482-9
Print ISBN: 978-1-4020-8481-2
DOI: https://doi.org/10.1007/978-1-4020-8482-9

Neuer Inhalt

Bildnachweise

VDI-Icon, Profil Icon, inhalt2, Springer Professional Modul/© Springer Fachmedien Wiesbaden GmbH, Die Gewinner und Laudatoren des Sustainability Award in Automotive 2024/© Uli Regenscheit | ATZlive, Search Icon, Banner Hanser, EU-Lieferkettengesetz/© Z6944 Sascha Steinach / ZB / picture alliance, Brems- und Reifenabrieb/© tomazl / Getty Images / iStock, Dr. Fabian Struck/© Forto Logistics SE & Co., Zeitschrift Wissensmanagement Cover, PatentFit-Logo/© Springer Fachmedien Wiesbaden GmbH, chassis.tech plus 2023/© [M] ATZlive / TÜV SÜD PRODUCT SERVICE GMBH, adäsion-Webinar-Matinee/© krystiannawrocki_ Getty Images, Lack- und Pulvertreff/© Spinger Fachverlage Wiesbaden GmbH

Springer Professional